For \(f \in \mathcal {S}(\mathbb {R}^{n+m})\) , let \(T_{\mu _U} f = \mu _U \ast f\)  and let \(\mathcal {R}f = \widehat{f} \, |_{\Sigma }\) ,
where \(\widehat{f}\)  denotes the usual Fourier transform of \(f\) . The \(L^{p}(\mathbb {R}^{n+m}) - L^{q}(\Sigma )\)  boundedness of the restriction
operator \(\mathcal {R}\)  to the sub-manifold \(\Sigma \subset \mathbb {R}^{n+m}\)  has been widely studied, in different cases, by many authors.
For \(S^{1} \subset \mathbb {R}^{2}\)  by C. Fefferman in [1]}, for \(S^{n-1} \subset \mathbb {R}^{n}\) , \(n \ge 2\) , by P. Tomas and
E. Stein in [2]} and [3]} respectively, for quadratic surfaces with non vanishing gaussian curvature by R. Strichartz in [4]}. The case of compact \(n-\) dimensional manifolds \(\Sigma \subset \mathbb {R}^{2n}\)  was studied by E. Prestini in [5]} under the assumption that: for each \(x_0 \in \Sigma \)  there exists a local chart \(X(x_1, ..., x_n)\)  around \(x_0\)  satisfying that the vectors
\(\left\lbrace  \displaystyle {\frac{\partial X}{\partial x_i}} \right\rbrace _{i=1}^{n}\)  and
\(\left\lbrace  \displaystyle {\frac{\partial ^{2} X}{\partial x_i \partial x_j}} \right\rbrace _{i, j=1}^{n}\)  span \(\mathbb {R}^{2n}\) .
The case of two dimensional manifolds \(\Sigma = \left\lbrace  (x, \varphi (x)) : x \in U \subset \mathbb {R}^{2}\right\rbrace  \subset \mathbb {R}^{4}\) 
was studied by M. Christ in [6]}, where a restriction theorem is given under the assumption that: for each \(x \in U\)  and
\(\theta \in \mathbb {R}\) , if \(\det \left( \left\langle \varphi ^{\prime \prime }(x), (\cos (\theta ), \sin (\theta )) \right\rangle \right) = 0\) 
(where \(\langle \, , \, \rangle \)  denotes the inner product in \(\mathbb {R}^{2}\) ) then
\(\frac{d}{d\theta }\det \left( \left\langle \varphi ^{\prime \prime }(x), (\cos (\theta ), \sin (\theta )) \right\rangle \right) \ne 0\) . More restriction theorems for homogeneous sub-manifolds of \(\mathbb {R}^{n}\)  can be found in [7]} and [8]}. A very interesting book about the Fourier restriction problem can be found in [9]}.
